Complex numbers are fundamental mathematical concepts with wide-ranging applications in science and engineering. Understanding them is essential for solving advanced problems in fields like physics, electrical engineering, and signal processing.

This article presents a variety of important complex number questions. Whether you’re a student or simply interested in expanding your mathematical knowledge, these questions and explanations will help you improve your problem-solving skills and grasp the power of complex numbers in real-world applications.

What are Complex Numbers?

Complex numbers are numbers of the form [Tex]a+ib[/Tex], where a is the real part and b is the imaginary part with i defined as [Tex]\sqrt{-1}[/Tex]​. Complex numbers extend the real numbers to solve equations that have no real solutions. Widely used in engineering, physics, and applied mathematics, complex numbers facilitate the analysis of signals and control systems.

Complex Numbers

Important Formulas: Complex Numbers

• Complex number representation: z = a + bi, where i² = -1
• Addition and subtraction: (a + bi) ± (c + di) = (a ± c) + (b ± d)i
• Multiplication: (a + bi)(c + di) = (ac – bd) + (ad + bc)i
• Division: (a + bi) / (c + di) = [(ac + bd) / (c² + d²)] + [(bc – ad) / (c² + d²)]i
• De Moivre’s Theorem: (r(cosθ + isinθ))ⁿ = rⁿ(cos(nθ) + isin(nθ))
• Euler’s Formula: e^(iθ) = cosθ + isinθ
• Polar form of a complex number: z = r(cosθ + isinθ), where r = |z| and θ = arg(z)
• Roots of unity: The nth roots of unity are given by: z_k = e^((2πki)/n), where k = 0, 1, 2, …, n-1

Complex Numbers Important Questions with Solutions

Question 1: Express the complex number 3 + 4i in polar form.

Solution:

Magnitude: r = √(3² + 4²) = √25 = 5

Angle: θ = arctan(4/3) ≈ 0.9273 radians or 53.13°

Polar form: 5(cos 0.9273 + i sin 0.9273) or 5e^(0.9273i)

Question 2: Find the roots of the equation z² + 4z + 13 = 0 in the complex plane.

Solution:

Using the quadratic formula: z = (-b ± √(b² – 4ac)) / 2a

z = (-4 ± √(16 – 52)) / 2 = (-4 ± √-36) / 2 = (-4 ± 6i) / 2

Roots: z₁ = -2 + 3i and z₂ = -2 – 3i

Question 3: Calculate (2 + 3i)(4 – 5i).

Solution:

(2 + 3i)(4 – 5i) = 8 – 10i + 12i – 15i² = 8 + 2i + 15 = 23 + 2i

Modulus and Conjugate of a Complex number

• Conjugate of a complex number: (a + bi)* = a – bi
• Modulus (absolute value) of a complex number: |z| = √(a² + b²)

Question 4: Find the modulus and argument of the complex number 1 + i.

Solution:

Modulus: |1 + i| = √(1² + 1²) = √2 ≈ 1.414

Argument: arg(1 + i) = arctan(1/1) = π/4 radians or 45°

Question 5: Evaluate e^(iπ/4).

Solution:

Using Euler’s formula: e^(iθ) = cos θ + i sin θ

e^(iπ/4) = cos(π/4) + i sin(π/4) = (√2/2) + i(√2/2)

Question 6: Find the complex conjugate of 3 – 2i.

Solution:

The complex conjugate is obtained by changing the sign of the imaginary part.

Complex conjugate of 3 – 2i = 3 + 2i

Question 7: Simplify (1 + i)⁴.

Solution:

(1 + i)⁴ = ((1 + i)²)² = (2i)² = -4

Question 8: Find the real and imaginary parts of (2 + 3i) / (1 – i).

Solution:

(2 + 3i) / (1 – i) = ((2 + 3i)(1 + i)) / ((1 – i)(1 + i))

= (2 + 2i + 3i – 3) / (1 + 1) = (-1 + 5i) / 2

Real part: -1/2, Imaginary part: 5/2

Question 9: Calculate the cube roots of -8.

Solution:

Cube roots of -8 = 2(cos((2k + 1)π/3) + i sin((2k + 1)π/3)), where k = 0, 1, 2

Root 1: 2(cos(π/3) + i sin(π/3)) = 1 + i√3

Root 2: 2(cos(π) + i sin(π)) = -2

Root 3: 2(cos(5π/3) + i sin(5π/3)) = 1 – i√3

Question 10: Find the distance between the complex numbers 3 + 4i and 1 – 2i in the complex plane.

Solution:

Distance = |3 + 4i – (1 – 2i)| = |2 + 6i| = √(2² + 6²) = √40 = 2√10

Question 11: Express the complex number i⁹⁹ in the form a + bi, where a and b are real numbers.

Solution:

i⁴ = 1, so i⁹⁹ = i⁽⁹⁶⁺³⁾ = i⁹⁶ × i³ = i^4×24 × i³ = (i⁴)²⁴ × (-i) = (1)²⁴ × (-i) = -i

Therefore, i⁹⁹ = 0 – i

Question 12: Find the value of sin(iπ/2).

Solution:

Using the formula sin(ix) = i sinh(x):

sin(iπ/2) = i sinh(π/2) ≈ i × 2.3012

Question 13: Solve the equation z⁴ = 16 in the complex plane.

Solution:

z = 16^(1/4) × (cos((2kπ)/4) + i sin((2kπ)/4)), where k = 0, 1, 2, 3

z¹ = 2(cos 0 + i sin 0) = 2

z² = 2(cos(π/2) + i sin(π/2)) = 2i

z³ = 2(cos π + i sin π) = -2

z⁴ = 2(cos(3π/2) + i sin(3π/2)) = -2i

Question 14: Calculate the integral of e^z around the unit circle in the complex plane.

Solution:

Using Cauchy’s integral formula: ∫e^z dz = 2πi × e⁰ = 2πi

Question 15: Find the real and imaginary parts of (cos θ + i sin θ)ⁿ.

Solution:

Using De Moivre’s formula: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ)

Real part: cos(nθ), Imaginary part: sin(nθ)

Question 16: Find the complex number z such that z + (1/z) = 2.

Solution:

Let z = x + yi. Then (x + yi) + (1/(x + yi)) = 2

Multiplying both sides by (x + yi): (x + yi)²+ 1 = 2(x + yi)

x²– y² + 2xyi + 1 = 2x + 2yi

Equating real and imaginary parts:

x² – y² + 1 = 2x and 2xy = 2y

From the second equation, x = 1 or y = 0

If y = 0, then x² + 1 = 2x, so x = 1

Therefore, z = 1 or z = -1

Question 17: Find the Laurent series of f(z) = 1/(z-1)(z-2) about z = 1.

Solution:

f(z) = 1/(z-1) – 1/(z-2) = 1/(z-1) – 1/(1+(z-1))

= 1/(z-1) – 1/(1-(-(z-1))) = 1/(z-1) + Σ(-(z-1))ⁿ for n=0 to ∞

Laurent series: 1/(z-1) + 1 – (z-1) + (z-1)² – (z-1)³ + …

Question 18: Prove that if |z| < 1, then |1 – z| > |1 + z|.

Solution:

|1 – z|² – |1 + z|² = (1 – z)(1 – z̄) – (1 + z)(1 + z̄)

= 1 – z – z̄ + |z|² – 1 – z – z̄ – |z|²

= -2(z + z̄) = -4Re(z)

Since |z| < 1, Re(z) < 1, so -4Re(z) > 0

Therefore, |1 – z|² > |1 + z|², and |1 – z| > |1 + z|

Read More,

• Complex Numbers
• Natural Numbers
• Real Numbers

Complex Numbers Practice Questions

Question 1: Evaluate the following complex expression: (3 + 4i)³

Question 2: Find all complex numbers z that satisfy the equation: z⁴ + 16 = 0

Question 3: If z = cos θ + i sin θ, express 1/z in terms of cos θ and sin θ.

Question 4: Prove that for any two complex numbers a and b, |a + b|² + |a – b|² = 2(|a|² + |b|²)

Question 5: Find the locus of points in the complex plane that satisfy the equation |z – 2| = |z + 2|

Complex Numbers Questions with Solutions – FAQs

What are complex Numbers?

Complex numbers are numbers of the form [Tex]a+ib[/Tex], where a and b are real numbers, and i is the imaginary unit with [Tex]i^2=−1.[/Tex] They are used to solve equations that have no real solutions and have applications in engineering, physics, and applied mathematics.

What are different forms to represent Complex numbers?

Complex numbers can be represented in several different forms:

1. Standard Form: [Tex]a+ib[/Tex], where a is the real part and b is the imaginary part.
2. Polar Form: r(cosθ+isinθ), where r is the magnitude and θ is the argument (angle).
3. Exponential Form: [Tex]re^{i\theta}[/Tex], which is a compact representation using Euler’s formula, relating to the polar form.